The effects of a strong confinement on the textures of liquid crystals have been thoroughly studied for various mesophases, such as nematics or cholesterics. But the confinement of smectics has been less studied, because the presence of parallel layers yields complex textures and macroscopic defects, commonly identified as focal conic domains1. Although it has been shown that any interface can be relaxed by focal conic domains2, the choice of a given texture by the material has not yet been clearly formalized. For example, direct smectic droplets confined in a sphere with a planar anchoring display numerous domains whereas inverse smectic droplets submitted to the same anchoring are much simpler and usually show only two singular lines arising from the surface (the equivalent of the boojums of inverse nematic droplets3). We have studied the topological and geometrical consequences of a strong anchoring at any tilt angleθ0 on a complex surface. We have introduced the notion of “STS” (smectic texture on a surface), in contrast with “STB” (smectic texture in the bulk), and we have classified the possible point defects on any surface . The bipolar STS is the simplest possible texture, but can be found on surfaces of revolution4 only. This analysis has been used to explain several generic textures of confined smectics, such as the typical size of the domains, the evolution of the textures with the zenithal anchoring angle θ0, the discrepancies in the textures of direct and inverse droplets…